Loss Coherence – Part I


What is Loss Coherence? Define the Loss Coherence Coefficient; Next post –  Explore Loss Coherence behavior and values.

I’m not a fan of correlation.

You can read that a couple of different ways. First, I’m not a fan of high correlations. Second, I’m not a fan of correlation coefficients in general, as they are often used in the financial industry. I’m addressing the second part in this post.

I could write a whole post on why I dislike correlation. Maybe another time. But briefly, let me point out something that has always bothered me.

Correlation measures the tendency of the month-to-month change in the returns of two investments to go in the same direction. In other words, if the returns for both investments are higher or lower by approximately the same amount this month than last month, that will tend to increase correlation. Conversely, if one investment has a higher return this month than last month and the other investment has a lower return this month than last month, that will tend to decrease correlation.

At no point in calculating correlation do we look at the returns of the two investments and ask whether either investment lost money, and more importantly, we never ask whether the returns of one investment tended to cancel out or reinforce losses in the other investment.

Instead, what if we ask the question, “What is the tendency of positive returns from one investment to cancel out negative returns from another investment? Conversely, what is the tendency for both of my investments to have negative returns during the same period?”

After all, isn’t that what we really want to know?

What is Loss Coherence?

Loss coherence is the tendency of two investments to have negative returns in the same period. In the context of a portfolio, when an investment loses money, we would like for another investment to make money to offset the loss. What we don’t want is for all the investments in our portfolio to lose money at the same time. So loss coherence is a bad thing, and the more of it we have, the worse our portfolio is going to perform when it comes to drawdowns.

Loss coherence is roughly analogous to correlation of returns, in that the more of it we have, the worse our portfolio will perform. Where it differs is that it measures a very specific kind of correlation, the tendency of two investments to have negative returns at the same time.

In the rest of this article, I will define the loss coherence coefficient {{C}_{L}} and begin to explore its behavior.

Expectation of Negative Returns for Two Investments

Consider the returns of two investments, InvA and InvB, over a period containing N samples in each time series. If {{n}_{A}} is the number of negative returns for InvA in the period and {{n}_{B}}  is the number of negative returns for InvB in the period, then, given that the returns are independent and randomly distributed, the probability of any given sample of InvA having a negative return is {{P}_{{A-}}}={{n}_{A}}/N and the probability of any given sample of InvB having a negative return is {{P}_{{B-}}}={{n}_{B}}/N. Further, the probability of both investments having negative returns in the same sample is

(1)   \begin{equation*}{{P}_{{AB-}}}={{P}_{{A-}}}{{P}_{{B-}}}\end{equation*}

Throughout this post,  we will use a test case for InvA and InvB where N=60{{n}_{A}}=30, and {{n}_{B}}=30. It’s trivial then to see that {P}_{{AB-}} is 0.25.

The Binomial Distribution

The binomial distribution refers to the probability of having k successful tests out of N total tests, where each test has a binary outcome (i.e., it’s either successful or it isn’t) and the probability of any given test being successful is p. It’s easy to calculate with the Excel Binom.Dist() function or any number of online calculators.

In the case of our two investments InvA and InvB, we have N=60 and p=0.25. The PDF and CDF for this case are shown in Chart 1.

 

Chart 1 – Binomial Distribution N=60, p=0.25

For this distribution, the mode is 15 and the probability of having 15 or fewer successful tests is 56.9%.

What if we examine the returns, however, and find there are actually 25 samples where both returns are negative? The probability of having 25 or more samples like this is only 0.3%. In this case, there are far more samples with both returns being negative than are predicted, so we can say that the losses are highly coherent. Since these two investments are likely to suffer losses at the same time, I might think twice before including both of them in a portfolio.

Conversely, what if there are only 8 samples where both returns are negative? The probability of seeing 8 or fewer occasions where both returns are negative is just 2.1%, so we can say that the loss coherence is very low, and including both of the investments in a portfolio would be a reasonable thing to do.

Quantifying Loss Coherence

How might we quantify the difference between the actual number and the expected number of samples where both returns are negative? Because loss coherence is roughly analogous to correlation, I’d like to have a function that looks like correlation. In other words, the function should have a minimum value of -1, a maximum value of +1, and if we apply it to two random variables, it should have a mean of 0.

For the binomial distribution with N tests and probability of success p, let \displaystyle F(k;N,p) denote the cumulative density function. The binomial distribution has a mode of:

(2)   \begin{equation*}$\displaystyle \text{M}=\left\lfloor {\left( {N+1} \right)p} \right\rfloor $\end{equation*}

when \left( {N+1} \right)p is not an integer and \left\lfloor{\left{X}\right}\rfloor\right denotes the floor of X.

When \left( {N+1} \right)p is an integer, there are two modes with identical probabilities:

(3)   \begin{equation*}$\displaystyle {{M}_{U}}=(N+1)p$\end{equation*}

(4)   \begin{equation*}$\displaystyle {{M}_{L}}=(N+1)p-1$\end{equation*}

My strategy for quantifying loss coherence is shown graphically in Chart 2.

If the number of periods k where both returns are negative is greater than or equal to M (or {M}_{U}), then C_L is the cumulative probability of k successful tests minus the probability of the mode, divided by one minus the probability of the mode. In other words, C_L is how far away from the probability of the mode the probability of k successful tests is.

Likewise, if k is less than M (equal to or less than M_L), then C_L is the probability of k minus the probability of the mode, divided by the probability of the mode.

Chart 2 – Strategy for calculating loss coherence.

To formalize the definition of the loss coherence function C_L, if (N+1)p is not an integer, then

(5)   \begin{equation*}$\displaystyle {{C}_{L}}=\frac{{F(k;N,p)-F(M;N,p)}}{{1-F(M;N,p)}}$\end{equation*}

when k\ge M, and

(6)   \begin{equation*}$\displaystyle {{C}_{L}}=\frac{{F(k;N,p)-F(M;N,p)}}{{F(M;N,p)}}$\end{equation*}

otherwise.

If (N+1)p is an integer, then

(7)   \begin{equation*}$\displaystyle {{C}_{L}}=\frac{{F(k;N,p)-F(M_U;N,p)}}{{1-F(M_U;N,p)}}$\end{equation*}

when k\ge M_U, and

(8)   \begin{equation*}$\displaystyle {{C}_{L}}=\frac{{F(k;N,p)-F(M_L;N,p)}}{{F(M_L;N,p)}}$\end{equation*}

otherwise.

Note that, like correlation, C_L has a minimum value of -1, a maximum value of +1, and a value of 0 when the actual number of samples with two negative returns is equal to the mode.

Referring back to Chart 2, if we find 21 samples where both returns are negative, then the loss coherence would be:

\displaystyle {{C}_{L}}=\frac{{94.6-56.9}}{{1-56.9}}=0.87,

which is a high loss coherence and might make them unsuitable for inclusion in the same portfolio. If instead we find only 9 samples where both returns are negative, then the loss coherence would be:

\displaystyle {{C}_{L}}=\frac{{4.5-56.9}}{{56.9}}=-0.92,

which is a very low loss coherence coefficient and would indicate they might be a good pair of investments to include in the portfolio.

Summary

Using the binomial distribution, we have defined a function that calculates the loss coherence coefficient. The loss coherence is roughly analogous to correlation of returns, except that it measures a very specific property, the tendency of two investments to lose money at the same time.

In the next post in this series, I’ll calculate the loss coherence for a number of different cases and begin to explore its behavior and how it might be useful.